Intuitive questions about the shape of an $\ell_1$ ball in dimensions $n \geq 4$?

Question

It is easy to intuitively see that the shape of a two-dimensional $\ell_1$ ball is a sort of diamond, and that the three-dimensional generalization of it will be a similar shape, i.e., a shape where if you slice into it along any of the planes orthogonal to the coordinate axes, you get the two-dimensional $\ell_1$ ball. Is this also true of a four dimensional $\ell_1$ ball? Can you slice it with the hyperplane $(1,0,0,0)^\perp$ and get the 3-dimensional ball? If not, how can we describe the shape of the four-dimensional ball, and the shape in higher dimensions? Is it exactly the regular $n$-dimensional polytope with vertices at the vectors $\pm e_i$? Any answers are appreciated, thank you

Answer 1

What you're looking for is the cross-polytope. In the plane this is a square, and in $\Bbb R^3$, it's an octahedron.